CERNTH/95180
Introduction
A year ago Seiberg and Witten [1] showed how to obtain an exact solution for the low energy effective action of supersymmetric YangMills theory with gauge group . Shortly afterwards the generalization to gauge groups with arbitrary has been obtained in [2, 3]. For these cases a lot of work has been done by now. The large limit and the connection to theories has been studied in [4]. The nonperturbative effective action has been elaborated in [5]. Phases with mutually nonlocal massless dyons were identified in [6]. Also supersymmetric QCD with matter in the fundamental representation of the gauge group has been considered in [7]. Fairly recently a solution for gauge groups has been presented in [8]. The purpose of this letter is to extend these previous results to gauge groups . It should be noted here that in the Cartan classification the groups correspond to simply laced Lie algebras of type whereas correspond to nonsimply laced ones of type .
Semiclassical Monodromies
Following [1] we write the Lagrangian for SYM with arbitrary gauge group in superspace language as
(1) 
where the vectormultiplet has been decomposed into an chiral superfield and an vectorfield . Two facts are of utmost importance here. Firstly the whole theory is governed by a single holomorphic function, the prepotential . Secondly there is a nontrivial classical potential for the scalar field, . From this it follows that for generic vacuum expectation value of the gauge group is broken down to the maximal torus. Thus the low energy degrees of freedom correspond to a gauge theory with being the rank of . Since can always be chosen to lie in the Cartan subalgebra, whose generators we denote by , we have . The Wbosons corresponding to the roots of the gauge group acquire a mass which is proportional to . Whenever the vacuum expectation value of the scalar field is orthogonal to a root, a Wboson becomes massless and therefore the low energy description is no longer valid there. This information can be compactly encoded in the zeroes of the Weyl group invariant “classical” discriminant:
(2) 
Here is the set of positive roots. Two vectors should be identified if they differ by a Weyl transformation of the gauge group. In the space of the the zeroes of the discriminant coincide with the fixed points under the Weyl group action.
Quantum mechanically the theory is characterized by a dynamically generated scale . It is weakly coupled if is large in comparison to , and therefore a perturbative description is valid in this regime. The prepotential takes the form [5, 8]
(3) 
It is clear that the logarithm gives rise to singularities if . is not a singlevalued function of , and thus gives rise to nontrivial monodromies when one encircles a singularity in the moduli space of vacuum configurations. Actually what one is interested in are monodromies acting on the vector where the dual variables are defined as . Thus for a gauge group of rank r the monodromy group will be a subgroup of the group of duality transformations .
In the perturbative regime there will be simple monodromies corresponding to the simple roots of the gauge group. They generate all the other semiclassical monodromies by conjugation. The monodromy matrix induced by the Weyl reflection on the root from the prepotential is
(4) 
To be specific we write down the simple monodromies for . We chose an orthogonal basis such that the four simple roots are given by [9]
(5) 
The four simple monodromies are
(6) 
According to the Dynkin diagram of , they fulfill the braid group relations
(7) 
with .
The Curves
Our aim is to reproduce these monodromies from an algebraic curve. There are several ways to arrive at the desired expression. First let us calculate where the matrices are in the fundamental representation of .
(8) 
We distinguished here the roots of the polynomial from the vacuum expectation values . This is because in the quantum case both can be identified only in the semiclassical regime. The Weyl group of is the semidirect product of . The first factor are permutations of elements and the second denotes simultaneous sign changes of two elements , . This distinguishes the series from the nonsimply laced case. In the latter one can flip the signs of each individually. The important consequence is that the series has an exceptional Casimir of order (cfg. (11) below). Clearly is invariant under the Weyl group. Therefore we can equally well write it in terms of gauge invariant variables
(9) 
with
(10)  
(11) 
Note that due to the structure of the Weyl group the polynomial depends quadratically on !
It has been suggested in [5] that the algebraic curves describing the moduli space of SYM with gauge groups based on the simply laced Lie algebras of ADEtype should be intimately related to the simple singularities of the same type. Indeed the curves are determined by the “LGpotentials” of type . Also the curves given in [8] are determined in exactly the same manner by the well known potentials for the boundary singularities [10]. How can we establish such a relationship for the case at hand?
The perturbed simple singularities for the series is given by
(12) 
Formally integrating out the variable by its equation of motion and then substituting one arrives at
(13) 
This is in fact wellknown in the context of conformal field theory [11]. Then our polynomial is simply . At this stage it is useful to have an explicit look at the discriminant of . To be specific, we chose the simplest generic case and change notation according to , ,
(14) 
Here is the discriminant that one computes from the LandauGinzburg potential (12) or from the onevariable expression (13). Using the basis of roots (5) one finds that it precisely reproduces (2). Thus our polynomial has an additional and a priori unexpected singularity at . We will see in the following how this puzzle resolves.
Following the standard arguments given in [2, 3], we suggest for the quantum theory the following family of algebraic curves
(15) 
The classical theory has a symmetry under which the field has charge . In the quantum theory it is anomalous and broken down to a discrete symmetry [1]. Here is the dual Coxeter number of the gauge group. In the case of it is given by . If we assign charge to and to , we see that our curve has precisely the required symmetry. The right hand side of (15) factorizes into
(16) 
From this we infer that similarly as for , the quantum discriminant factorizes as well. We see that has the form of the classical polynomial if we substitute . The factors have common roots only at a point in the moduli space where is a root. However this case is already contained in itself and therefore no new singularity arises. Thus the quantum discriminant is given by
(17) 
Strong and weak coupling monodromies
An important feature of our curve is that there appear only even
powers of . Therefore it is invariant under a symmetry
acting on the complex plane . This implies
that the homology can be uniquely decomposed into two subspaces
where is the space of cycles which are
invariant under and is the antiinvariant subspace
[10]. Our choice of cycles is depicted in Fig.1.
One sees that the cycles belong to whereas . From the latter we define antiinvariant cycles by taking the differences
(18) 
Cycles of this form are called long, cycles of the form of the are called short. Their intersection form is given by
(19) 
In order to define the variables and their duals, , as period integrals, we also need a suitable meromorphic oneform . For curves of the form (15) it has already been derived in [8]. In our case it is given by
(20) 
Since it also changes sign under the action of , we obtain invariant periods only by integrating over antiinvariant cycles. Thus the physically relevant subspace of the homology is . Now the fields can be defined as
(21) 
An additional subtlety arises in considering the PicardLefschetz formula. Since the intersection of any long antiinvariant cycle with another antiinvariant cycle is always even we have to correct this by a factor . We obtain the modified PicardLefschetz formula [10]
(22)  
(23) 
where is a long antiinvariant vanishing cycle and a short one. It should be noted here that in the case of we will never encounter short antiinvariant vanishing cycles. This is no surprise since these cycles correspond in a onetoone manner to short roots of Liealgebras. Also we could have chosen a basis of cycles consisting only of long antiinvariant ones. We did not do so because the choice in Fig.1 is best suited to the orthogonal basis of the Lie algebra. This will become clear from the discussion of the strong coupling monodromies. In addition, it allows most easily to compare and to work out the differences to the case when the gauge group is .
In order to derive the semiclassical monodromies
(S0.Ex1,6), we use the fact that they can
be written as a product of two strong coupling monodromies. More
precisely, we have chosen a slice through the moduli space by fixing
, , and varying only . Away from intersections or cusps
of the singular lines one sees four pairs of singular lines. Fixing a
base point in this plane we traced the effect on the branch points
when looping around the singularities. Depending on the particular
form of the loops, one finds different vanishing cycles. It suffices
however to choose eight loops, each encircling different singular
lines.
All the other monodromies can then be obtained by conjugation from the eight basic ones. The eight vanishing cycles can be seen in Fig. 2. Of physical relevance are again the differences . They can be expanded in terms of and . We find
The first entries refer to magnetic quantum numbers and the others to electric ones. A general formula for the monodromy matrix for a massless dyon of magnetic charge and electric charge has been given in [2]:
(24) 
Note that the magnetic quantum numbers of are given by the root , that of and that of by and respectively and finally that of by the root . According to this we reproduce the weak coupling monodromies in the following manner:
Another nontrivial check of the curve (15) concerns the quantum shift matrix . It is obtained in the following way. Performing the rotation , one computes from the prepotential (3) the monodromy
(25) 
In the orthogonal basis we have . On the complex curve this rotation induces pure braidings of the branch cuts. A single braid of the cut induces . From the fact that appears to the power of two times the dual Coxeter number we see that the number of braiding for each branch cut is . Thus we find the quantum shift matrix in the orthogonal basis.
The Monodromy around ?
Now we come to a subtle and crucial point in our discussion. From the expression of the quantum discriminant (17) one expects that some BPS state becomes massless as we approach for generic values of , and . A look at the algebraic curve (15) tells us that four branch points i.e. two branch cuts will collide at in this case, which seems to be a nasty, unstable situation.
But what does really happen as we loop around ? We have checked this situation for several onedimensional slices through the moduli space and in Fig. 3 we have depicted what happens to the branchpoints. Six of the branch cuts stay where they are, whereas two such cuts rotate around each other and end up at their original positions. Since no additional braidings occur, we see that the basic cycles are unchanged and the associated monodromy matrix is simply the identity. There are no massless particles associated with the “singularity around ” !
This is in strong contrast to what is happening for the nonsimply laced gauge groups . In that case the monodromy around ^{1}^{1}1By we mean the highest order gauge invariant operator of , which is of order has a massless particle associated with it and one finds a strong coupling monodromy. To be more specific we want to discuss in some detail. We use basic cycles analogous to Fig. 1 with being (they are the same as in [8]). The algebraic curve and the quantum discriminant read as follows:
(26) 
(27) 
In Fig. 4 we have drawn two antiinvariant vanishing cycles and associated with a short root.
For the vanishing cycle we can calculate the monodromy in the usual way and the magneticelectric quantum numbers read . Some difficulty arises for , which corresponds to a loop around ^{2}^{2}2 is the highest order Casimir for SO(5) in the moduli space. In this case the two branch points encircled by the vanishing cycle are not simply exchanged, but they rotate around and move back to their starting points, picking up a nontrivial braid. The monodromy obtained by using the PicardLefschetz formula (23) is not the correct one, because it corresponds to the exchange of the two branch points. In fact one has to take its square with quantum numbers i.e. the effective vanishing cycle is therefore . As expected, one obtains a weak coupling monodromy by multiplying two strong coupling monodromies. is precisely the monodromy matrix of [8] and corresponds to a Weyl reflection on a short root ! In the case of the singularity around is essential to account for the presence of short roots. Of course, short roots are not present for .
The Vacua
Now we want to take a closer look at the moduli space of gauge theories and the singularities living in it. The vanishing of the quantum discriminant (17)^{3}^{3}3in the following we will suppress the factor in (17) defines an dimensional singular submanifold in our dimensional moduli space. Since the quantum discriminant factorizes into two pieces, one expects to find at least two branches of the singular submanifold that intersect in some dimensional manifold. Furthermore, the branches split themselves into several branches and have self intersections and singularities, like cusps. For arbitrary one will find a discrete set of points where branches intersect simultaneously. In the following we will concentrate on these singular points, but it should be mentioned that there are also higher dimensional singular submanifolds which would be well worth to study.
At these singular points we can choose local coordinates and the quantum discriminant will factorize into factors (to lowest order). If some of the factors are linear dependent, we are at a cusp singularity where several of the massless dyons are mutually nonlocal. Such points have been studied for the case of in [4]. At some special points all branches intersect transversely i.e. the factors are all linearly independent. All massless particles are mutually local, and one can break supersymmetry down to by adding a mass term.
We have studied the case of in some detail, and found 48 points where four branches of the singular manifold intersect. Mutually nonlocal dyons are present at 42 of these points, whereas the remaining six points are the vacua
(28) 
with l = 0 …5. This is the correct number of vacua for the gauge theory. We also checked this for and found vacua.
For larger it becomes increasingly difficult to find the vacua directly from the quantum discriminant. We propose therefore a different approach which is similar to the discussion in [4]. If we insert the values (28) into our curve (15) we find a polynomial with two simple roots, two double roots and one sixfold root at . This situation can easily be generalized for arbitrary with the help of Chebyshev polynomials. The classical polynomial (9) takes the form
(29) 
whereas the curve (15) will be of the form
(30) 
We obtain a total of solutions by complex rotation which is the correct number of ground states. As a check one can compare the expansion of (29) for with the classical curve (9), and read off the values of , , and . They perfectly agree with the solutions found directly from the examination of the singular points of the quantum discriminant of given in (17). We want to point out that the vacuum points found for the series, with , coincide with the vacua for the , series with all odd Casimirs set to zero.
In Fig. 5 we have depicted how the branch cuts arrange and
what cycles vanish as we approach an vacuum point. The
vanishing cycles have the usual antiinvariant form . Apparently and are
non intersecting. For and the situation
is not so obvious but the antiinvariant combinations
and turn out to have vanishing
intersection form. Thus we find four mutually local dyons as it
should be for a vacuum.
Finally we note that in the last figure one sees a mirror pair of the Dynkin diagram. This and the appearance of the Chebyshev polynomials is obviously related to the resolution of simple singularities [10]. It also allows for another beautiful interpretation. Identifying the electric cycles around the branch cuts with weights and the magnetic vanishing cycles with simple roots we recognize the weight diagram of the fundamental representation of !
Acknowledgements
We would like to thank W. Lerche for drawing our attention to this problem and for many helpful discussions. This work has been supported by the BMWFK.
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